Suppose I am in location X, emitting a sound at some loudness level. Bob, standing at location Y, hears the sound with an intensity reduced by $x$ db. If Bob starts emitting sound, will I also hear it at an intensity reduced by exactly $x$ db? (In general, there may be some general configuration of doors/walls between me and Bob, not symmetric between the two of us. These barriers might absorb some sound energy, as well as reflecting, refracting and/or diffracting the sound waves).
Now, at a technical level one can make arguments that the answer is yes; for example, the propagation of sound waves at some frequency should be described by the Helmholtz equation $$(\nabla^2 + k(x)^2) \varphi = 0,$$ where $k(x)$ is the (possibly spatially-dependent) inverse wavelength of sound at location $x$. (We can also make $k(x)$ complex to include the possibility of dissipation). The intensity of sound at location $y$ emitted from position $x$ is then described by the Green's function $G(x,y)$ of the Helmholtz equation, which can be shown to be symmetric.
However, this argument seems way too technical and doesn't really answer the basic question of why the propagation of sound waves had to be described by a PDE with symmetric Green's function. Are there any exotic scenarios where this symmetry property does not hold? Does it depend on any of the approximations made in deriving the Helmholtz equation? Or is there some basic physical argument for why the symmetry property must be true?
To anticipate a possible answer, I do not believe this has anything to do with time reversal symmetry. For one thing, time reversal symmetry does not hold in the presence of absorption. And in any case, the time reverse of Bob emitting a sound is a bunch of sound waves converging on Bob, not me emitting a sound.